# How to describe the kernel of a homomorphism

I am generally unsure of what a kernel is in abstract algebra. From my background in linear algebra, I understand that the kernel is the same as the null space (I learned it as the null space).

For example, if I have some homomorphism $$f: (\mathbb{R},+) \longrightarrow (\mathbb{C}^*, \cdot)$$, how would I go about describing the kernel?

• Kernel is the set of elements of the domain which are mapped to the identity element of the co-domain. Commented Nov 12, 2020 at 7:40

## 1 Answer

Ther kernel of a (group) homomorphism is the set of all elements that get mapped onto the neutral element.

So $$\ker(f)=\{x\in\mathbb{R}: f(x)=1\}$$, as $$1$$ is the neutral element of $$\mathbb{C}^\ast$$ with regards to multiplication.

How to describe the kernel then depends clearly on the specific homomorphism.

• So any element that maps to the identity element in the range? Thank you so much! Commented Nov 12, 2020 at 7:47
• Yes. Also the kernel is never empty, as every homomorphism sends the identity element onto the identity element. But as I said, how to discribe the kernel depends on the homomorphism. As a tip: You should always clearify definitions and notation with your lecture notes. Commented Nov 12, 2020 at 7:51